Fractales: La Belleza De La Naturaleza

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FRACTALES: LA BELLEZA DE LA NATURALEZA (y su relación con la pintura expresionista) 1/12 J. Palacián y C. Martínez (UPNa e IES Alhama) B. Mandelbrot #1 B. Mandelbrot (1967): “How long is the British coastline?” 2/12 J. Palacián y C. Martínez (UPNa e IES Alhama) B. Mandelbrot #2 Es el principal exponente del interés por la Geometría fractal. Mostró cómo los fractales aparecen en muchos campos, tanto en las Matemáticas como, sobre todo, en la Naturaleza. Fractal viene del latín fractus, que significa roto o fracturado. 3/12 J. Palacián y C. Martínez (UPNa e IES Alhama) 4/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Características de los Fractales 1. Estructura que se repite en escalas cada vez más pequeñas. 2. Es demasiado irregular para ser descrita por la Geometría Euclídea. 3. Estructura geométrica que es dividida en partes, cada una de las cuales es (al menos aproximadamente) una copia de tamaño reducido de la estructura original. 4. Se forma por iteración: La definición es recursiva. 5/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Dimensión fractal #3 Dimensión fractal Tenemos un objeto para el que necesitamos ensamblar N copias para construir una versión más grande con un factor de escala S. La dimensión fractal del objeto se define como el número real positivo d, que cumple: Sd=N 6/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Ejemplo: Curva de Koch #1 7/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Ejemplo: Curva de Koch #2 • ¿Cuántas copias de la curva original son necesarias para construir una versión más grande? 4. • ¿Cuál es el factor de escala que hay que aplicar a la Curva de Koch para obtener la curva más grande inmediatamente posterior? 3 (La longitud del segmento “que hace el mismo papel” en una versión y en la posterior está multiplicada por 1/3) Así: 3d=4, entonces, d=log(4)/log(3)=1.26185..... 8/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Fractales en la pintura #1 Jackson Pollock fue un artista estadounidense y un referente del expresionismo abstracto. Suyas son varias pinturas que se encuentran entre las más caras del mundo. Una de ellas es la Number 5,1948 que se vendió en 148 millones de dólares. 9/12 J. Palacián y C. Martínez (UPNa e IES Alhama) 10/12 J. Palacián y C. Martínez (UPNa e IES Alhama) 11/12 J. Palacián y C. Martínez (UPNa e IES Alhama) Fractales en la pintura #7 “Las pinturas de Jackson Pollock tienen una dimensión fractal característica, a dos escalas, que se acentúa con los años” (Richard Taylor) ¿Podemos nosotros pintar como Jackson Pollock? ¡SI! J. Palacián y C. Martínez (UPNa) 12/58 Programas para generar fractales Fractal Fractal Caos Pro Explorer Forge ChaoScope Ultra fractal Winfract Mandelbulb Mandelbox Apophysis 3D (2009) (2010) Fraktal Fractalus Studio Fractint 13/12 J. Palacián y C. Martínez (UPNa e IES Alhama) .

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